I am happy with the way you characterize my beliefs in the first paragraph of your blog. Unemployment is clearly not Pareto optimal. Everything you say after that is at best misleading and at worst dismissive of everything we (at least some of us) learned from Keynes.
The idea of involuntary unemployment was introduced by Keynes in the General Theory. But you already knew that. It is defined as a situation where (in modern language) the ratio of the marginal disutility of work to the marginal utility of consumption is not equal to the real wage. That seems a pretty accurate description of the equilibrium outcome of labor search models.
Bob Lucas cast a spell over the profession in a series of papers in the 1970s. You are accurately summarizing Bob's view. That view was tied to a three decade long campaign by economists predominately located in Chicago, Minnesota and Rochester (at the time) to discredit Keynesian economics. Tom Sargent reputedly advised his students not to read the General Theory. That was a tragic mistake and we are still suffering from the consequences.
You are right to assert that the important distinction is between equilibria that are Pareto optimal and those that are not. You are wrong to assert that the term 'involuntary unemployment' has no useful meaning.
I accept your categorization of the allocation of time between three competing ends. Every family, and every member of that family, chooses every day whether they will choose to participate in the labor force. As long as they are in the labor force, they may be employed or unemployed. Those who are unemployed do not choose that state. They must wait for a job offer to appear. In some states, that job offer may take a couple of days to arrive. In others, it may take a couple of years. The activity of waiting for a job, even when it involves active search, can meaningfully be called involuntary unemployment.
The dismissal of 'involuntary unemployment' from the lexicon of the modern economist was introduced as part of a deliberate attack on Keynesian economics. It is time to roll back that attack. As I have shown here, 'involuntary' unemployment is a useful way of distinguishing unemployment that is part of a social optimum, from unemployment that is not.